35361 Probability and Stochastic Processes, Spring 2015

Assignment 1 (due to September 9, 2015)

The assignment should be handed in to Tim Ling on 9/09/2015. You

may use any software for calculations (Mathematica is preferable).

Problem 1.

1) Let Xi, i = 1, 2 be independent random variables (i.r.v.’s) having a

Chi-square distribution, EXi = 4.

(i) Using characteristic functions and the inversion formula find the

probability density function (pdf) of Y = X1 – X2.

(ii) Find E(Y 8).

1

Problem 2. 1) Using a variance reduction technique (e.g. control

variates) find the Monte-Carlo approximation for the integral

J=

Z 8

0

2

-x

e

(1 + x2)dx.

Use the sample sizes n = 106, n = 107 and compare the results with

the exact value.

2) Using the 3-sigma rule estimate a sample size n required for obtaining a Monte-Carlo approximation with a control variate for J with

an absolute error less than ? = 10-6.

2

Problem 3.

Let B0(t), t ? [0, 1] be a Brownian Bridge that is a Gaussian process

with E(B0(t)) = 0 and the covariance function

R(t, s) = min(t, s) – ts.

1) Using simulations with a discrete-time process approximation for

B0(t) (e.g. use N=1000 trajectories and n=1000 discretisation points)

find an approximation for the distribution function of the random variable

X = max |B0(t)|

0=t=1

at the points {0.2, 0.6, 2.0}. Hint: use a representation for B0(t) in

terms of a standard Brownian motion.

2) Verify the results using the analytical expression for the distribution

function of X :

P {X < x} = 1 + 2

8

X

2 x2

k

-2k

(-1) e

.

k=1

3